Ok, figured it out. Thankfully on my remote machine I didn’t install any addons or change the user preferences, so I just ran the following
blender -b --python-console # Start python console
>> import bpy
>> bpy.ops.wm.read_factory_settings() # Load factory settings
>> bpy.ops.wm.save_userpref() # Save factory settings
I was then able to open the python console again and simply set the CUDA device and graphics card using the method from this thread like I did before: Thread
After doing this the script I used displayed the following (correct) results:
CUDA
AMD FX(tm)-6350 Six-Core Processor
P106-100
And GPU rendering worked perfectly.